In the following, we present numerical results and compare the proposed algorithm with JOTPA suggested in [1]} and AB algorithm.
The pseudo-code of the proposed HBCT algorithm is summarized in Algorithm . It is worth mentioning that \(\textbf {A} = [A_1, \dots , A_K]\) . As proved in Proposition REF , to maximize end-to-end bit delivery, the number of transmission bits for all users should be equal. The problem is solved by considering Proposition REF ; hence, substituting optimal solutions into (REF ) gives us maximum end-to-end bit delivery for a given \(\mathbf {\lambda }\) . To find the optimal solution for \(\mathbf {\lambda }\) , we adopt the fminunc tool in MATLAB.
We assume that the length of each time block is \(\text{T} = 1\) .
The channel power gains are calculated as \(Y_k = \vert \beta _k^Y \vert ^2(\frac{d_k^Y}{d_0})^{-\alpha }(Y=h, g, f)\) . We assume fading channels have independent Rayleigh fading; hence, \(\vert \beta _k^Y \vert ^ 2\)  is an independent exponential random variable. \(d_k^Y\)  is the link distance, \(d_0\)  is the reference distance, and \(\alpha \)  is the path-loss exponent. Similar to those in [1]}, [3]}, [4]}, [5]}, the value of the path-loss exponent is considered \(\alpha = 2\) , which is for free space environments. Herein, we assume that the reference distance is \(d_0 = 1\) . The number of hops is considered \(K=3\)  unless mentioned else. We assume that energy harvesting efficiency in each \(SU_k\)  is \(\zeta = 0.8\)  [1]}, [7]}. The noise power in received signals at each \(SU_k\)  is assumed to be \(\sigma ^2 = 1\) . Therefore, the powers of all devices are normalized to \(\sigma ^2\) . The power of signals emitted by the PT and the tolerable interference in PR are \(P_t=40 dB\)  and \(I_p = 0 dB\) , respectively. The location of PT and PR is (-8,10) and (-2,10), respectively. The location of the source node, i.e., \(SU_1\) , and the destination node, i.e., the \(SU_K\) , is assumed to be \((-10,0)\)  and \((0,0)\) , respectively. We assume that the intermediate nodes are located in a straight line from \(SU_1\)  to the \(SU_K\) . The bitrate of backscatter transmission is summarized in Table REF . These values are obtained from a practical experiment presented in [8]}.
<FIGURE>